↑

Properties

Label 18.0.95772567332...4375.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{9}\cdot 7^{14}\cdot 67^{14}$
Root discriminant $463.05$
Ramified primes $3, 5, 7, 67$
Class number $2469903744$ (GRH)
Class group $[2, 2, 2, 4, 77184492]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![305535379456, -2937968128, 87416317952, 16619785184, 11737837080, 1825666152, 2370420196, 151200629, -19380689, 38273266, 25199154, -1533003, -1363905, 18035, 30077, 42, -282, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 282*x^16 + 42*x^15 + 30077*x^14 + 18035*x^13 - 1363905*x^12 - 1533003*x^11 + 25199154*x^10 + 38273266*x^9 - 19380689*x^8 + 151200629*x^7 + 2370420196*x^6 + 1825666152*x^5 + 11737837080*x^4 + 16619785184*x^3 + 87416317952*x^2 - 2937968128*x + 305535379456)
 
gp: K = bnfinit(x^18 - x^17 - 282*x^16 + 42*x^15 + 30077*x^14 + 18035*x^13 - 1363905*x^12 - 1533003*x^11 + 25199154*x^10 + 38273266*x^9 - 19380689*x^8 + 151200629*x^7 + 2370420196*x^6 + 1825666152*x^5 + 11737837080*x^4 + 16619785184*x^3 + 87416317952*x^2 - 2937968128*x + 305535379456, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 282 x^{16} + 42 x^{15} + 30077 x^{14} + 18035 x^{13} - 1363905 x^{12} - 1533003 x^{11} + 25199154 x^{10} + 38273266 x^{9} - 19380689 x^{8} + 151200629 x^{7} + 2370420196 x^{6} + 1825666152 x^{5} + 11737837080 x^{4} + 16619785184 x^{3} + 87416317952 x^{2} - 2937968128 x + 305535379456 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-957725673320066238974132504381157911244615234375=-\,3^{9}\cdot 5^{9}\cdot 7^{14}\cdot 67^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $463.05$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 7, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{5}{32} a^{6} - \frac{1}{8} a^{5} - \frac{7}{32} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} - \frac{5}{32} a^{7} - \frac{1}{8} a^{6} - \frac{7}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{11136} a^{14} + \frac{49}{11136} a^{13} - \frac{169}{11136} a^{12} - \frac{143}{3712} a^{11} + \frac{23}{464} a^{10} - \frac{223}{2784} a^{9} - \frac{445}{11136} a^{8} + \frac{1879}{11136} a^{7} - \frac{801}{3712} a^{6} + \frac{51}{3712} a^{5} - \frac{271}{1392} a^{4} - \frac{117}{464} a^{3} - \frac{229}{1392} a^{2} + \frac{37}{174} a - \frac{40}{87}$, $\frac{1}{33408} a^{15} - \frac{1}{33408} a^{14} - \frac{61}{11136} a^{13} + \frac{365}{33408} a^{12} - \frac{73}{1856} a^{11} - \frac{337}{8352} a^{10} + \frac{1003}{33408} a^{9} + \frac{619}{11136} a^{8} - \frac{8309}{33408} a^{7} - \frac{499}{11136} a^{6} - \frac{1603}{16704} a^{5} + \frac{73}{522} a^{4} + \frac{269}{4176} a^{3} + \frac{131}{2088} a^{2} - \frac{95}{261} a + \frac{86}{261}$, $\frac{1}{143534398464} a^{16} + \frac{10705}{23922399744} a^{15} - \frac{1426717}{35883599616} a^{14} + \frac{214419667}{71767199232} a^{13} + \frac{1788537719}{143534398464} a^{12} + \frac{923009495}{35883599616} a^{11} - \frac{23559515}{5316088832} a^{10} + \frac{849562091}{71767199232} a^{9} - \frac{986612951}{8970899904} a^{8} + \frac{8691040997}{71767199232} a^{7} - \frac{14594829611}{143534398464} a^{6} + \frac{188487523}{4485449952} a^{5} - \frac{1085020333}{8970899904} a^{4} - \frac{1933869359}{5980599936} a^{3} + \frac{17593523}{1121362488} a^{2} - \frac{2382275}{62297916} a + \frac{140041517}{280340622}$, $\frac{1}{42129213793097018941068431702860037843442193177497109378191910287017812992} a^{17} + \frac{113670672939281589449806492358963535728924329218649335491243723}{42129213793097018941068431702860037843442193177497109378191910287017812992} a^{16} - \frac{92548055995246531942818623958522388074620232481203967162039568613263}{21064606896548509470534215851430018921721096588748554689095955143508906496} a^{15} - \frac{274507024077050188365369807305201114202352725222943022750077563106297}{7021535632182836490178071950476672973907032196249518229698651714502968832} a^{14} + \frac{218872809051593157614506993840049571497583123187099352385122999086195855}{14043071264365672980356143900953345947814064392499036459397303429005937664} a^{13} + \frac{1404709211716458129287682341995669582407201354062694894055062732163567}{4681023754788557660118714633651115315938021464166345486465767809668645888} a^{12} + \frac{2530374340416768884469968959251004676461767611489335396320864646356750115}{42129213793097018941068431702860037843442193177497109378191910287017812992} a^{11} - \frac{687049439400392336282713818972767625491419552999241390166327519014801447}{42129213793097018941068431702860037843442193177497109378191910287017812992} a^{10} + \frac{353647735161624194646108566940173173457269405036374415083572817762850753}{7021535632182836490178071950476672973907032196249518229698651714502968832} a^{9} + \frac{710404506535063690104550560987264419063619942810650816574798680177852963}{7021535632182836490178071950476672973907032196249518229698651714502968832} a^{8} + \frac{3372038710829757485229784479431325352991762321701255360178737412499878117}{14043071264365672980356143900953345947814064392499036459397303429005937664} a^{7} + \frac{1124805933686944276944432971566067162553015268500937026312119151542180017}{42129213793097018941068431702860037843442193177497109378191910287017812992} a^{6} - \frac{978452755898914871963970830130490667342264775353228691603879268499881}{5044206632315256099265856286261977711140109336386148153519146346625696} a^{5} + \frac{284271926499362509664250638659603824348983331122908840270862324205612521}{5266151724137127367633553962857504730430274147187138672273988785877226624} a^{4} - \frac{2594206978509798675357468233123635236503310414033838399491250000170393385}{5266151724137127367633553962857504730430274147187138672273988785877226624} a^{3} - \frac{14004540862272847739206912388706132447308454040806040358984938935820876}{41141810344821307559637140334824255706486516774899520877140537389665833} a^{2} + \frac{23297556703431104477175246223607973095258227757477496896462561652823291}{164567241379285230238548561339297022825946067099598083508562149558663332} a - \frac{10171225882464558636568629985688444032145231990303811843582850629154703}{82283620689642615119274280669648511412973033549799041754281074779331666}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{77184492}$, which has order $2469903744$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4641319478.753615 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.3.219961.2, 3.3.469.1, 6.0.742368375.2, 6.0.163292090133375.1, 9.9.4991256617582519389.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$67$67.6.4.2$x^{6} - 67 x^{3} + 53868$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.12.10.3$x^{12} - 16147 x^{6} + 93083904$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$